We present a framework based on convex optimization and spectral regularization to perform learning when feature observations are multidimensional arrays (tensors). We give a mathematical characterization of spectral penalties for tensors and analyze a unifying class of convex optimization problems for which we present a provably convergent and scalable template algorithm. We then specialize this class of problems to perform learning both in a transductive as well as in an inductive setting. In the transductive case one has an input data tensor with missing features and, possibly, a partially observed matrix of labels. The goal is to both infer the missing input features as well as predict the missing labels. For induction, the goal is to determine a model for each learning task to be used for out of sample prediction. Each training pair consists of a multidimensional array and a set of labels each of which corresponding to related but distinct tasks. In either case the proposed technique exploits precise low multilinear rank assumptions over unknown multidimensional arrays; regularization is based on composite spectral penalties and connects to the concept of Multilinear Singular Value Decomposition. As a by-product of using a tensor-based formalism, our approach allows one to tackle the multi-task case in a natural way. Empirical studies demonstrate the merits of the proposed methods.Editor: Massimiliano Pontil.

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Time-optimal path following considers the problem of moving along a predetermined geometric path in minimum time. In the case of a robotic manipulator with simplified constraints a convex reformulation of this optimal control problem has been derived previously. However, many applications in robotics feature constraints such as velocity-dependent torque constraints or torque rate constraints that destroy the convexity. The present paper proposes an efficient sequential convex programming (SCP) approach to solve the corresponding nonconvex optimal control problems by writing the non-convex constraints as a difference of convex (DC) functions, resulting in convex-concave constraints. We consider seven practical applications that fit into the proposed framework even when mutually combined, illustrating the flexibility and practicality of the proposed framework. Furthermore, numerical simulations for some typical applications illustrate the fast convergence of the proposed method in only a few SCP iterations, confirming the efficiency of the proposed framework.

Source separation or demixing is the process of extracting multiple components entangled within a signal. Contemporary signal processing presents a host of difficult source separation problems, from interference cancellation to background subtraction, blind deconvolution, and even dictionary learning. Despite the recent progress in each of these applications, advances in high-throughput sensor technology place demixing algorithms under pressure to accommodate extremely high-dimensional signals, separate an ever larger number of sources, and cope with more sophisticated signal and mixing models. These difficulties are exacerbated by the need for real-time action in automated decision-making systems.Recent advances in convex optimization provide a simple framework for efficiently solving numerous difficult demixing problems. This article provides an overview of the emerging field, explains the theory that governs the underlying procedures, and surveys algorithms that solve them efficiently. We aim to equip practitioners with a toolkit for constructing their own demixing algorithms that work, as well as concrete intuition for why they work. Fundamentals of demixingThe most basic model for mixed signals is a superposition model, where we observe a mixed signal z 0 ∈ R d of the formand we wish to determine the component signals x 0 and y 0 . This simple model appears in many guises. Sometimes, superimposed signals come from basic laws of nature. The amplitudes of electromagnetic waves, for example, sum together at a receiver, making the superposition model (1) common in wireless communications. Similarly, the additivity of sound waves makes superposition models natural in speech and audio processing.Other times, a superposition provides a useful, if not literally true, model for more complicated nonlinear phenomena. Images, for example, can be modeled as the sum of constituent featuresthink of stars and galaxies that sum to create an image of a piece of the night sky [1]. In machine learning, superpositions can describe hidden structure [2], while in statistics, superpositions can model gross corruptions to data [3]. These models also appear in texture repair [4], graph clustering [5], and line-spectral estimation [6].A conceptual understanding of demixing in all of these applications rests on two key ideas. Low-dimensional structures: Natural signals in high dimensions often cluster around lowdimensional structures with few degrees of freedom relative to the ambient dimension [7].Examples include bandlimited signals, array observations from seismic sources, and natural

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